<!DOCTYPE html>
<!--********************************************-->
<!--*       Generated from PreTeXt source      *-->
<!--*                                          *-->
<!--*         https://pretextbook.org          *-->
<!--*                                          *-->
<!--********************************************-->
<html lang="en-US">
<head xmlns:og="http://ogp.me/ns#" xmlns:book="https://ogp.me/ns/book#">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Eigenvalue Problems</title>
<meta name="Keywords" content="Authored in PreTeXt">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta property="og:type" content="book">
<meta property="book:title" content="MATH 2023: Ordinary and Partial Differential Equations">
<meta property="book:author" content="Xiaoyi Chen and Wei Zhang">
<script src="https://sagecell.sagemath.org/static/embedded_sagecell.js"></script><script>window.MathJax = {
  tex: {
    inlineMath: [['\\(','\\)']],
    tags: "none",
    tagSide: "right",
    tagIndent: ".8em",
    packages: {'[+]': ['base', 'extpfeil', 'ams', 'amscd', 'newcommand', 'knowl']}
  },
  options: {
    ignoreHtmlClass: "tex2jax_ignore|ignore-math",
    processHtmlClass: "process-math",
    renderActions: {
        findScript: [10, function (doc) {
            document.querySelectorAll('script[type^="math/tex"]').forEach(function(node) {
                var display = !!node.type.match(/; *mode=display/);
                var math = new doc.options.MathItem(node.textContent, doc.inputJax[0], display);
                var text = document.createTextNode('');
                node.parentNode.replaceChild(text, node);
                math.start = {node: text, delim: '', n: 0};
                math.end = {node: text, delim: '', n: 0};
                doc.math.push(math);
            });
        }, '']
    },
  },
  chtml: {
    scale: 0.88,
    mtextInheritFont: true
  },
  loader: {
    load: ['input/asciimath', '[tex]/extpfeil', '[tex]/amscd', '[tex]/newcommand', '[pretext]/mathjaxknowl3.js'],
    paths: {pretext: "https://pretextbook.org/js/lib"},
  },
startup: {
    pageReady() {
      return MathJax.startup.defaultPageReady().then(function () {
      console.log("in ready function");
      $(document).trigger("runestone:mathjax-ready");
      }
    )}
},
};
</script><script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script><script src="https://pretextbook.org/js/lib/jquery.min.js"></script><script src="https://pretextbook.org/js/lib/jquery.sticky.js"></script><script src="https://pretextbook.org/js/lib/jquery.espy.min.js"></script><script src="https://pretextbook.org/js/0.13/pretext.js"></script><script>miniversion=0.674</script><script src="https://pretextbook.org/js/0.13/pretext_add_on.js?x=1"></script><script src="https://pretextbook.org/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
</script><link href="https://fonts.googleapis.com/css?family=Open+Sans:400,400italic,600,600italic" rel="stylesheet" type="text/css">
<link href="https://fonts.googleapis.com/css?family=Inconsolata:400,700&amp;subset=latin,latin-ext" rel="stylesheet" type="text/css">
<link href="https://pretextbook.org/css/0.4/pretext.css" rel="stylesheet" type="text/css">
<link href="https://pretextbook.org/css/0.4/pretext_add_on.css" rel="stylesheet" type="text/css">
<link href="https://pretextbook.org/css/0.4/banner_default.css" rel="stylesheet" type="text/css">
<link href="https://pretextbook.org/css/0.4/toc_default.css" rel="stylesheet" type="text/css">
<link href="https://pretextbook.org/css/0.4/knowls_default.css" rel="stylesheet" type="text/css">
<link href="https://pretextbook.org/css/0.4/style_default.css" rel="stylesheet" type="text/css">
<link href="https://pretextbook.org/css/0.4/colors_blue_red.css" rel="stylesheet" type="text/css">
<link href="https://pretextbook.org/css/0.4/setcolors.css" rel="stylesheet" type="text/css">
</head>
<body class="pretext-book ignore-math has-toc has-sidebar-left">
<a class="assistive" href="#content">Skip to main content</a><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\lt}{&lt;}
\newcommand{\gt}{&gt;}
\newcommand{\amp}{&amp;}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle     \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle        \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle      \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)</span></div>
<header id="masthead" class="smallbuttons"><div class="banner"><div class="container">
<a id="logo-link" href=""></a><div class="title-container">
<h1 class="heading"><a href="MATH-2023-OPDE.html"><span class="title">MATH 2023: Ordinary and Partial Differential Equations</span></a></h1>
<p class="byline">Xiaoyi Chen and Wei Zhang</p>
</div>
</div></div>
<nav id="primary-navbar" class="navbar"><div class="container">
<div class="navbar-top-buttons">
<button class="sidebar-left-toggle-button button active" aria-label="Show or hide table of contents sidebar">Contents</button><div class="tree-nav toolbar toolbar-divisor-3">
<a class="index-button toolbar-item button" href="index-1.html" title="Index">Index</a><span class="threebuttons"><a id="previousbutton" class="previous-button toolbar-item button" href="sec7_1.html" title="Previous">Prev</a><a id="upbutton" class="up-button button toolbar-item" href="ch_seven.html" title="Up">Up</a><a id="nextbutton" class="next-button button toolbar-item" href="sec7_3.html" title="Next">Next</a></span>
</div>
</div>
<div class="navbar-bottom-buttons toolbar toolbar-divisor-4">
<button class="sidebar-left-toggle-button button toolbar-item active">Contents</button><a class="previous-button toolbar-item button" href="sec7_1.html" title="Previous">Prev</a><a class="up-button button toolbar-item" href="ch_seven.html" title="Up">Up</a><a class="next-button button toolbar-item" href="sec7_3.html" title="Next">Next</a>
</div>
</div></nav></header><div class="page">
<div id="sidebar-left" class="sidebar" role="navigation"><div class="sidebar-content">
<nav id="toc"><ul>
<li class="link frontmatter"><a href="meta_frontmatter.html" data-scroll="meta_frontmatter" class="internal"><span class="title">Front Matter</span></a></li>
<li class="link">
<a href="ch_first.html" data-scroll="ch_first" class="internal"><span class="codenumber">1</span> <span class="title">Introduction</span></a><ul>
<li><a href="sec_1-intro.html" data-scroll="sec_1-intro" class="internal">Classification of Differential Equations</a></li>
<li><a href="sec_2-intro.html" data-scroll="sec_2-intro" class="internal">Linear and Nonlinear Equation</a></li>
<li><a href="sec_3-intro.html" data-scroll="sec_3-intro" class="internal">Geometrical Aspect</a></li>
<li><a href="sec_4-intro.html" data-scroll="sec_4-intro" class="internal">Motivation</a></li>
</ul>
</li>
<li class="link">
<a href="ch_second.html" data-scroll="ch_second" class="internal"><span class="codenumber">2</span> <span class="title">First Order Ordinary Differential Equations</span></a><ul>
<li><a href="sec2_1.html" data-scroll="sec2_1" class="internal">Linear Equations</a></li>
<li><a href="sec2_2.html" data-scroll="sec2_2" class="internal">Further Discussion of Linear Equations (For reading only)</a></li>
<li><a href="sec2_3.html" data-scroll="sec2_3" class="internal">Separable Equations</a></li>
<li><a href="sec2_4.html" data-scroll="sec2_4" class="internal">Difference Between Linear and Nonlinear Equations</a></li>
<li><a href="sec2_5.html" data-scroll="sec2_5" class="internal">Applications of modeling with first order ODE(For reading only)</a></li>
<li><a href="sec2_6.html" data-scroll="sec2_6" class="internal">Exact Equations and Integrating Factors</a></li>
</ul>
</li>
<li class="link">
<a href="ch_third.html" data-scroll="ch_third" class="internal"><span class="codenumber">3</span> <span class="title">third Order Linear Equations</span></a><ul>
<li><a href="sec3_1.html" data-scroll="sec3_1" class="internal">Homogeneous equations with constant coefficient</a></li>
<li><a href="sec3_2.html" data-scroll="sec3_2" class="internal">Fundamental Solutions of Linear Homogeneous Equations</a></li>
<li><a href="sec3_3.html" data-scroll="sec3_3" class="internal">Linear Independence and Wronskian</a></li>
<li><a href="sec3_4.html" data-scroll="sec3_4" class="internal">Complex roots of the characteristic equations</a></li>
<li><a href="sec3_5.html" data-scroll="sec3_5" class="internal">Repeated Roots: Reduction of Order</a></li>
<li><a href="sec3_6.html" data-scroll="sec3_6" class="internal">Non-homogeneous Equations and Method of Undetermined Coefficients</a></li>
<li><a href="sec3_7.html" data-scroll="sec3_7" class="internal">Variation of Parameters</a></li>
</ul>
</li>
<li class="link">
<a href="ch_four.html" data-scroll="ch_four" class="internal"><span class="codenumber">4</span> <span class="title">Higher Order Linear Equations</span></a><ul>
<li><a href="sec4_1.html" data-scroll="sec4_1" class="internal">General Theory of the <span class="process-math">\(n\)</span>-th Order Linear Equations</a></li>
<li><a href="sec4_2.html" data-scroll="sec4_2" class="internal">Homogeneous Equations with Constant Coefficients</a></li>
<li><a href="sec4_3.html" data-scroll="sec4_3" class="internal">The Method of Undetermined Coefficients</a></li>
<li><a href="sec4_4.html" data-scroll="sec4_4" class="internal">The Method of Variation of Parameters</a></li>
</ul>
</li>
<li class="link">
<a href="ch_five.html" data-scroll="ch_five" class="internal"><span class="codenumber">5</span> <span class="title">Series Solutions of Second Order Linear Equations</span></a><ul>
<li><a href="sec5_1.html" data-scroll="sec5_1" class="internal">Brief Review on Power Series</a></li>
<li><a href="sec5_2.html" data-scroll="sec5_2" class="internal">Introduction</a></li>
<li><a href="sec5_3.html" data-scroll="sec5_3" class="internal">Series Solutions Near an Ordinary Point</a></li>
<li><a href="sec5_4.html" data-scroll="sec5_4" class="internal">Euler’s Equation</a></li>
<li><a href="sec5_5.html" data-scroll="sec5_5" class="internal">Series Solution near a Regular Singular Point</a></li>
</ul>
</li>
<li class="link">
<a href="ch_six.html" data-scroll="ch_six" class="internal"><span class="codenumber">6</span> <span class="title">System of First Order Linear Equations</span></a><ul>
<li><a href="sec6_1.html" data-scroll="sec6_1" class="internal">Introduction <span class="process-math">\(\&amp;\)</span> Basic Theory</a></li>
<li><a href="sec6_2.html" data-scroll="sec6_2" class="internal">Homogeneous System with Constant Coefficients</a></li>
<li><a href="sec6_3.html" data-scroll="sec6_3" class="internal">Complex Eigenvalues</a></li>
<li><a href="sec6_4.html" data-scroll="sec6_4" class="internal">Repeated Eigenvalues</a></li>
<li><a href="sec6_5.html" data-scroll="sec6_5" class="internal">Fundamental Matrices</a></li>
<li><a href="sec6_6.html" data-scroll="sec6_6" class="internal">Non-homogeneous linear systems</a></li>
</ul>
</li>
<li class="link">
<a href="ch_seven.html" data-scroll="ch_seven" class="internal"><span class="codenumber">7</span> <span class="title">Partial Differential Equations</span></a><ul>
<li><a href="sec7_1.html" data-scroll="sec7_1" class="internal">Two-Point Boundary Value Problems</a></li>
<li><a href="sec7_2.html" data-scroll="sec7_2" class="active">Eigenvalue Problems</a></li>
<li><a href="sec7_3.html" data-scroll="sec7_3" class="internal">Fourier Series</a></li>
<li><a href="sec7_4.html" data-scroll="sec7_4" class="internal">The Fourier Convergence Theorem</a></li>
<li><a href="sec7_5.html" data-scroll="sec7_5" class="internal">Even and Odd Functions</a></li>
<li><a href="sec7_6.html" data-scroll="sec7_6" class="internal">Introduction to Partial Differential Equations</a></li>
<li><a href="sec7_7.html" data-scroll="sec7_7" class="internal">1D Heat Equation; Solutions by Separation of Variable and Fourier Series</a></li>
<li><a href="sec7_8.html" data-scroll="sec7_8" class="internal">Other Heat Conduction Problems</a></li>
</ul>
</li>
<li class="link">
<a href="ch_eight.html" data-scroll="ch_eight" class="internal"><span class="codenumber">8</span> <span class="title">Laplace transform</span></a><ul>
<li><a href="sec8_1.html" data-scroll="sec8_1" class="internal">What are Laplace Transforms, and Why?</a></li>
<li><a href="sec8_2.html" data-scroll="sec8_2" class="internal">Finding Laplace Transforms</a></li>
<li><a href="sec8_3.html" data-scroll="sec8_3" class="internal">Finding inverse transforms using partial fractions</a></li>
<li><a href="sec8_4.html" data-scroll="sec8_4" class="internal">Solving ODEs and ODE Systems</a></li>
<li><a href="sec8_5.html" data-scroll="sec8_5" class="internal">Step input and Impulse problems</a></li>
<li><a href="sec8_6.html" data-scroll="sec8_6" class="internal">Laplace transform for PDE (heat equation)</a></li>
</ul>
</li>
<li class="link">
<a href="ch_features.html" data-scroll="ch_features" class="internal"><span class="codenumber">9</span> <span class="title">Examples of PreTeXt features</span></a><ul><li><a href="sec_features-blocks.html" data-scroll="sec_features-blocks" class="internal">Environments and Blocks</a></li></ul>
</li>
<li class="link backmatter"><a href="meta_backmatter.html" data-scroll="meta_backmatter" class="internal"><span class="title">Backmatter</span></a></li>
<li class="link"><a href="solutions-1.html" data-scroll="solutions-1" class="internal"><span class="codenumber">A</span> <span class="title">Selected Hints</span></a></li>
<li class="link"><a href="solutions-2.html" data-scroll="solutions-2" class="internal"><span class="codenumber">B</span> <span class="title">Selected Solutions</span></a></li>
<li class="link"><a href="appendix-1.html" data-scroll="appendix-1" class="internal"><span class="codenumber">C</span> <span class="title">List of Symbols</span></a></li>
<li class="link"><a href="index-1.html" data-scroll="index-1" class="internal"><span class="title">Index</span></a></li>
<li class="link"><a href="colophon-1.html" data-scroll="colophon-1" class="internal"><span class="title">Colophon</span></a></li>
</ul></nav><div class="extras"><nav><a class="pretext-link" href="https://pretextbook.org">Authored in PreTeXt</a><a href="https://www.mathjax.org"><img title="Powered by MathJax" src="https://www.mathjax.org/badge/badge.gif" alt="Powered by MathJax"></a></nav></div>
</div></div>
<main class="main"><div id="content" class="pretext-content"><section class="section" id="sec7_2"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">7.2</span> <span class="title">Eigenvalue Problems</span>
</h2>
<p id="p-312">Consider the eigenvalue problem</p>
<div class="displaymath process-math" data-contains-math-knowls="" id="ep1">
\begin{equation}
y^{\prime \prime}+ \lambda y=0,\quad y(0)=0,\quad y(L)=0,\tag{7.2.1}
\end{equation}
</div>
<p class="continuation">We are interested in</p>
<ul id="p-313" class="disc">
<li id="li-25"><p id="p-314">non-trivial solutions   (<span class="process-math">\(y\equiv 0\)</span> is not considered since it is a trivial solution),</p></li>
<li id="li-26"><p id="p-315">and possible values of <span class="process-math">\(\lambda\)</span> that lead to non-trivial solutions.</p></li>
</ul>
<p id="p-316">Note that it is important to have <em class="emphasis">homogeneous</em> BCs for eigenvalue problems! And the eigenvalue problem is a two-point BVP. Depending on the BC, it might or might not have nontrivial solutions.</p>
<p id="p-317"><dfn class="terminology">Definition</dfn>  For some <span class="process-math">\({\lambda_n}\text{,}\)</span> if we are able to find a nontrivial solution <span class="process-math">\({y_n(x)}\text{,}\)</span> then, such <span class="process-math">\({\lambda_n}\)</span> is called an <dfn class="terminology">eigenvalue</dfn>, and <span class="process-math">\({y_n}\)</span> is the corresponding <dfn class="terminology">eigenfunction</dfn>.</p>
<p id="p-318">This type of problems is an important building block in series solutions of partial differential equations.(Dirichlet BC) We now attempt to solve the problem in <a href="" class="xref" data-knowl="./knowl/ep1.html" title="Equation 7.2.1">(7.2.1)</a>.</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/ep1.html">
\begin{equation*}
y^{\prime \prime}+ \lambda y=0,\quad y(0)=0,\quad y(L)=0,
\end{equation*}
</div>
<p class="continuation">The general solution depends on the roots, i.e., on the sign of <span class="process-math">\(\lambda\text{,}\)</span> which has 3 possibilities:</p>
<ol id="p-319" class="decimal">
<li id="li-27">
<p id="p-320">If <span class="process-math">\(\lambda&lt;0\text{,}\)</span> we write <span class="process-math">\(\lambda=-k^2\text{,}\)</span> where <span class="process-math">\(k=\sqrt{|\lambda|}&gt;0\text{,}\)</span>Then the characteristic equation yields <span class="process-math">\(r^2 =k^2~\to~r=\pm k\)</span> and the general solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y(x)=c_1e^{kx} + c_2e^{-kx}.
\end{equation*}
</div>
<p class="continuation">By boundary conditions, we must have</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
c_1 + c_2 = 0,~~ c_1e^{kL} + c_2e^{-kL} = 0,
\end{equation*}
</div>
<p class="continuation">which gives the solution <span class="process-math">\(c_1 = c_2 = 0\text{.}\)</span> Then <span class="process-math">\(y(x) = 0\text{,}\)</span> which is a trivial solution. Discard it.</p>
</li>
<li id="li-28">
<p id="p-321">If <span class="process-math">\(\lambda=0\text{,}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y'' = 0\quad \to\quad y(x)=Ax+B, ~~A, B\in\mathbb{R}.
\end{equation*}
</div>
<p class="continuation">With homogeneous boundary conditions, <span class="process-math">\(y(x) = 0\text{,}\)</span> therefore trivial. We also discard it.</p>
</li>
<li id="li-29">
<p id="p-322">If <span class="process-math">\(\lambda&gt;0\text{,}\)</span> we write <span class="process-math">\(\lambda=k^2\text{,}\)</span> where <span class="process-math">\(k=\sqrt{|\lambda|}&gt;0\text{,}\)</span> then <span class="process-math">\(r^2 =-k^2~\to~r=\pm ki\text{,}\)</span> and</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y(x)=c_1\cos {kx} + c_2\sin {kx}.
\end{equation*}
</div>
<p class="continuation">By setting in boundary conditions:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\left\{\begin{array}{l} y(0)={c_1~=0}\\
y(L)=c_2\sin kL=0\end{array}\right.
\end{equation*}
</div>
<p class="continuation">So it must be <span class="process-math">\(c_2\neq 0\)</span> and <span class="process-math">\(\sin kL = 0\)</span> leads to</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
kL=n\pi,\quad \Rightarrow \quad k=\frac{n\pi}{L},~~n=1,2,3,\cdots
\end{equation*}
</div>
<p class="continuation">We have found a family (infinite size) of eigenvalues and eigenfunctions! Using <span class="process-math">\(n\)</span> as the index, they are</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\lambda_n}={\left(\frac{n\pi}{L}\right)^2},\quad {y_n(x)}={\sin\frac{n\pi x}{L}},\quad n={1},2,3,\cdots
\end{equation*}
</div>
</li>
</ol>
<p id="p-323">In the next example, we will change the boundary conditions to another type, and we usually call it the Neumann type boundary conditions.</p>
<p id="p-324">(Neumann BC) Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y^{\prime \prime}+ \lambda y=0,\quad {y'(0)=0,\quad y'(L)=0},
\end{equation*}
</div>
<p id="p-325">To solve this problem, we still consider the same 3 cases of <span class="process-math">\(\lambda\text{:}\)</span> Denote <span class="process-math">\(k=\sqrt{|\lambda|}\text{,}\)</span></p>
<ol id="p-326" class="decimal">
<li id="li-30">
<p id="p-327">If <span class="process-math">\(\lambda=-k^2&lt;0\text{,}\)</span> then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y(x)=c_1e^{kx} + c_2e^{-kx},\quad y'(x)=kc_1e^{kx} - kc_2e^{-kx}
\end{equation*}
</div>
<p class="continuation">To determine <span class="process-math">\(c_1\text{,}\)</span> <span class="process-math">\(c_2\text{,}\)</span> use the boundary conditions:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
kc_1-kc_2=0,\quad kc_1e^{kL}-kc_2e^{-kL}=0,\quad\Rightarrow\quad c_1=c_2=0
\end{equation*}
</div>
<p class="continuation">which gives only trivial solution (Discard).</p>
</li>
<li id="li-31">
<p id="p-328">If</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y(x)=Ax+B,\quad\to\quad y'(x)=A
\end{equation*}
</div>
<p class="continuation">By boundary conditions, we have <span class="process-math">\(A=0,~B\in\mathbb{R}\text{.}\)</span> So we found an eigenpair:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\lambda_0=0,\qquad y_0(x)=1}.
\end{equation*}
</div>
</li>
<li id="li-32">
<p id="p-329">If <span class="process-math">\(\lambda=k^2&gt;0\text{,}\)</span> then <span class="process-math">\(r^2 =-k^2~\to~r=\pm ki\text{,}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y(x)=c_1\cos {kx} + c_2\sin {kx},\quad y'(x)=-kc_1\sin kx + kc_2\cos kx.
\end{equation*}
</div>
<p class="continuation">We now check the boundary conditions:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\left\{\begin{array}{l} y'(0)=kc_2=0\\
y'(L)=-kc_1\sin kL=0\end{array}\right.\Rightarrow \left\{\begin{array}{l} c_2=0\\
c_1\sin kL=0\end{array}\right.
\end{equation*}
</div>
<p class="continuation">If <span class="process-math">\(c_1 = 0\text{,}\)</span> then <span class="process-math">\(y(x) \equiv 0\)</span> which is trivial. So <span class="process-math">\(c_1\neq 0\)</span> and we must have <span class="process-math">\(\sin kL=0\text{:}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
kL=n\pi,\quad \Rightarrow \quad k=\frac{n\pi}{L},~~n=1,2,3,\cdots
\end{equation*}
</div>
<p class="continuation">For each <span class="process-math">\(k\text{,}\)</span> we get a pair of eigenvalue and eigenfunction</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\lambda_n}={\left(\frac{n\pi}{L}\right)^2},\quad {y_n(x)}={\cos\frac{n\pi x}{L}},\quad n={1},2,3,\cdots
\end{equation*}
</div>
</li>
</ol>
<p id="p-330">One could combine the results in 2 and 3, and get</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\lambda_n}={\left(\frac{n\pi}{L}\right)^2},\quad {y_n(x)}={\cos\frac{n\pi x}{L}},\quad n=\textcolor{red}{0},1,2,3,\cdots
\end{equation*}
</div>
<p class="continuation">Note that these are also a part of the trig set used in Fourier series! We will go to that part in the section <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "sec7_3.ptx" missing or not unique]</code>.</p>
<p id="p-331">Except for Dirichlet type and Neumann type of boundary conditions, an eigenvalue problem can also adopt boundary conditions of mixed type, or sometimes even more complicated one.</p>
<p id="p-332">(Mixed BC) Find all positive eigenvalues and their corresponding eigenfunctions of the problem</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y^{\prime \prime}+ \lambda y=0,\quad {y(0)=0,\quad y'(L)=0}.
\end{equation*}
</div>
<p id="p-333">If <span class="process-math">\(\lambda&gt;0\text{,}\)</span> we write <span class="process-math">\(\lambda = k^2\)</span> where <span class="process-math">\(k &gt; 0\text{,}\)</span> and the general solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y(x)=c_1\cos {kx} + c_2\sin {kx},\quad y'(x)=-kc_1\sin kx + kc_2\cos kx.
\end{equation*}
</div>
<p class="continuation">We now check the boundary conditions.</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\left\{\begin{array}{l} y(0)=c_1=0\\
y'(L)=kc_2\cos kL=0\end{array}\right.\Rightarrow \left\{\begin{array}{l} c_1=0\\
c_2\cos kL=0\end{array}\right.
\end{equation*}
</div>
<p class="continuation">which implies</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
kL=(n-\frac{1}{2})\pi,\quad \Rightarrow \quad k=\frac{\pi(n-\frac{1}{2})}{L},~~n=1,2,3,\cdots
\end{equation*}
</div>
<p class="continuation">We get the eigenvalues <span class="process-math">\(\lambda_n\)</span> and the corresponding eigenfunction <span class="process-math">\(y_n\)</span> as</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\lambda_n=\left(\frac{\pi(n-\frac{1}{2})}{L}\right)^2,~y_n=\sin \frac{\pi(n-\frac{1}{2})x}{L},~n=1,2,3,\cdots
\end{equation*}
</div>
<p id="p-334">For many other different boundary conditions, please refer to the textbook.</p>
<ul id="p-335" class="disc">
<li id="li-33"><p id="p-336">Different types of boundary conditions would give very different eigenvalues and eigenfunctions.</p></li>
<li id="li-34"><p id="p-337">In these examples, the eigenfunctions are sine and cosine functions, in the same form as the trig set we use in Fourier series. Recall that the trig set is a mutually orthogonal set. So, for each of these eigenvalue problems, the set of eigenfunctions are mutually orthogonal. In fact, this is a more general property for eigenfunctions. One can define proper inner product such that eigenfunctions for the same eigenvalue problem would always form a mutually orthogonal set.</p></li>
</ul></section></div></main>
</div>
</body>
</html>
